Answer
Local minimum of $f(2,-1)=-6$
Work Step by Step
Since, we have $f_x(x,y)=4x+3y-5=0, f_y(x,y)=3x+8y+2=0$
After solving the above two equations, we get
$x=2,y=-1$
Thus the critical point is: $(2,-1)$
As per second derivative test, we have
$D=f_{xx}f_{yy}-f^2_{xy}=23$ and $D=23 \gt 0$ and $f_{xx}=4 \gt 0$
Thus, local minimum of $f(2,-1)=-6$
